derivative of (sinx - cosx)/(sinx + cosx)

To find the derivative of the function \(\frac{\sin x - \cos x}{\sin x + \cos x}\), we'll use the quotient rule. The quotient rule states that if you have a function of the form \(\frac{u(x)}{v(x)}\), its derivative is given by:

\[
\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}
\]

Here, let:
- \( u(x) = \sin x - \cos x \)
- \( v(x) = \sin x + \cos x \)

### Step 1: Find the derivatives of \(u(x)\) and \(v(x)\)
- \( u'(x) = \cos x + \sin x \)
- \( v'(x) = \cos x - \sin x \)

### Step 2: Apply the quotient rule
Now, substitute \(u(x)\), \(v(x)\), \(u'(x)\), and \(v'(x)\) into the quotient rule formula:

\[
\frac{d}{dx}\left(\frac{\sin x - \cos x}{\sin x + \cos x}\right) = \frac{(\cos x + \sin x)(\sin x + \cos x) - (\sin x - \cos x)(\cos x - \sin x)}{(\sin x + \cos x)^2}
\]

### Step 3: Simplify the numerator
Let's expand and simplify the numerator:

1. Expand \((\cos x + \sin x)(\sin x + \cos x)\):
   \[
   (\cos x + \sin x)(\sin x + \cos x) = \cos^2 x + \cos x \sin x + \sin x \cos x + \sin^2 x = \cos^2 x + \sin^2 x + 2\cos x \sin x
   \]
   Since \(\cos^2 x + \sin^2 x = 1\), the expression simplifies to:
   \[
   1 + 2\cos x \sin x
   \]

2. Expand \((\sin x - \cos x)(\cos x - \sin x)\):
   \[
   (\sin x - \cos x)(\cos x - \sin x) = \sin^2 x - \sin x \cos x - \cos x \sin x + \cos^2 x = \sin^2 x + \cos^2 x - 2\sin x \cos x
   \]
   Again, \(\sin^2 x + \cos^2 x = 1\), so this simplifies to:
   \[
   1 - 2\sin x \cos x
   \]

Thus, the numerator becomes:
\[
(1 + 2\cos x \sin x) - (1 - 2\sin x \cos x) = 4\sin x \cos x
\]

### Step 4: Simplify the expression
Finally, the derivative is:
\[
\frac{4\sin x \cos x}{(\sin x + \cos x)^2}
\]

This can also be written using the identity \(\sin 2x = 2\sin x \cos x\) as:
\[
\frac{2\sin 2x}{(\sin x + \cos x)^2}
\]

### Final Answer
The derivative of \(\frac{\sin x - \cos x}{\sin x + \cos x}\) is:

\[
\frac{2\sin 2x}{(\sin x + \cos x)^2}
\]

Would you like any further details or explanations?

---

Here are 5 related questions:
1. What is the derivative of \(\frac{\sin x + \cos x}{\sin x - \cos x}\)?
2. How would you find the second derivative of \(\frac{\sin x - \cos x}{\sin x + \cos x}\)?
3. Can the derivative \(\frac{2\sin 2x}{(\sin x + \cos x)^2}\) be simplified further?
4. What is the integral of \(\frac{\sin x - \cos x}{\sin x + \cos x}\)?
5. How do you apply the quotient rule to functions involving trigonometric identities?

**Tip:** Remember that trigonometric identities can often simplify the differentiation or integration process, making complex expressions easier to handle.

Learn how to find the derivative of the function (sinx - cosx)/(sinx + cosx) using the quotient rule and trigonometric identities. This problem involves differentiation of trigonometric functions and applying the quotient rule formula. Suitable for advanced high school or college-level students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation